Time scale Batchelor

This expression was derived originally by Batchelor (1959) under the assumption that the correlation time of the Kolmogorov-scale strain rate is large compared with the Kolmogorov time scale. Alternatively, Kraichnan (1968) derived a model spectrum of the form... 

Batchelor (1959) developed an expression for the smallest concentration (or temperature) striation based on the argument that for diffusion time scales longer than the Kolmogorov scale, turbulence would continue to deform and stretch the blobs to smaller and smaller lamellae. Only once the lamellae could diffuse at the same rate as the viscous dissipation scale would the concentration striations disappear. The Batchelor length scale is the size of the smallest blob that can diffuse by molecular diffusion in one Kolmogorov time scale. Using the lamellar diffusion time from eq. (13-6) gives... 

Message From the comparisons above, it is clear that xm and tb scale-up in the same way. It turns out that each is proportional to l/N as long as geometry and the power per unit volume remain constant and the contribution due to the Batchelor scale mixing can be neglected. The key time scales for this problem are summarized in Table 13-2. Chemical reactions and their rates are scale-independent phenomena while the local mixing time is both scale and position dependent. Mixing effects get worse on scale-up. 

Vector fields whose divergence vanishes are sometimes referred to as solenoidal. A more comprehensive discussion of the conditions for approximating the velocity field as solenoidal has been given by Batchelor.8 These imply that, in cases in which the fluid is subjected to an oscillating pressure, the characteristic velocity in the Mach number condition should be interpreted as the product of the frequency times the linear dimension of the fluid domain, and that the difference in static pressures over the length scale of the domain must be small compared with the absolute pressure. Because our subject matter will frequently deal with incompressible, isothermal fluids, we shall often make use of (2-20) in lieu of the... 

In the previous sections we considered flows with a smooth spatial structure in which the relative dispersion of fluid trajectories is exponential in time and can be characterized by a single timescale, the inverse of the Lyapunov exponent. This is also valid for two-dimensional turbulent flows that have a smooth velocity field in the small-scale enstrophy cascade range (Bennett, 1984). A similar behavior occurs in any dimension at scales below the Kolmogorov scale (the so-called Batchelor or viscous-convective range, see below). In the inertial range of fully developed three-dimensional turbulence, however, the velocity field has a broad range of timescales and they all contribute to the relative dispersion of particle trajectories and affect the transport properties of the flow. 

This is the Batchelor spectrum, that is valid in the viscous-convective range that extends from the Kolmogorov scale down to the diffusive scale, where the scalar variance is finally dissipated by molecular diffusion. The diffusive scale in this case is the length scale at which the diffusion time l2/D is comparable to the timescale of advection corresponding to the Kolmogorov scale eddies, that gives... 

Mixing time determination should require a complete model with full description of velocity and concentration fields in the mixer. This is the difficult task of reactive flow simulation that would necessitate the description of transport, stretching and diffusion coupled with reaction of very fine structures down to the Batchelor scale... 

Spectral Arguments for Scalar Mixing and Mass Transfer. Batchelor (1959) used scaling argnments to determine the size of a pure sphere of dye that will diffuse in exactly the time it takes the energy in an eddy of size p to dissipate. This is called the Batchelor scale ... 

Figure 2-13 shows the gross characteristics of the velocity and concentration spectra. For a low viscosity liquid. Sc can be on the order of 1000, so the Batchelor scale can be 30 times smaller than the Kolmogorov scale. The ultimate scale of mixing needed for reaction is the size of a molecule, so in liquid-phase reactions, molecular diffusion is critically important for the final reduction in scale. For a gas, Sc is closer to 1, so the ratio is closer to 1, and the competition between the turbulent reduction in scale and molecular diffusion occurs at the same range of wavenumbers. The various length scales shown in Figure 2-13 are also summarized in Table 2-3... 

We could calculate the bulk blend time in the lab and in the plant, but in this case the process result requires a reaction. The reaction kinetics and molecular diffusivity are constant on scale-up, so we must ensure that the Batchelor scale is also preserved. The Batchelor scale can be defined using an estimate for the dissipation ... 

Batchelor length scale, (vD /e) " (m) absolute viscosity (kg/m s) kinematic viscosity (m /s) density (kg/m ) mixing time constant (s)... 

The inertial mixing (first term) is controlling since its time constant is about eight times the time constant of the Batchelor scale mixing. The Corrsin scale mixing is much faster than the reaction time constant. 

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